生物统计学答案 第十一章 多元回归及复相关分析.doc

第十一章 多元回归及复相关分析11.1 嗜酸乳杆菌（Lactobacillus acidophilus Lakcid） 是存在于肠道中的一种重要益生菌，为研究肠道中的条件对该菌生存的影响，设计了在体外不同的胆汁盐浓度和不同时间该菌的存活数（活菌数/mL），结果如下表59：时间/h胆汁盐/（g kg-1）123417.201081.041091.761092.041096.401068.401062.621031.7410321.641091.921099.601087.401081.221079.201062.091031.8910331.301091.421093.461086.001082.261062.041061.861031.8210349.801087.801081.021083.821081.301061.261061.321031.22103以该菌的存活数为因变量，胆汁盐浓度和时间为自变量，求二元回归方程并检验偏回归系数的显著性。答：程序和结果如下：options linesize=76 nodate;data mulreg;infile e:dataer11-1e.dat;input num time bile ;run;proc reg;model num=time bile;run;The SAS SystemThe REG ProcedureModel: MODEL1Dependent Variable: numAnalysis of VarianceSum of MeanSource DF Squares Square F Value Pr FModel 2 9.070013E18 4.535006E18 27.66 |t|Intercept 1 2020493645 237390215 8.51 .0001time 1 -144947822 64019380 -2.26 0.0312bile 1 -453586204 64019380 -7.09 .0001由以上结果得出回归方程：其中：X1为时间，X2为胆汁盐浓度。从偏回归系数的t检验结果可以得知，时间在0.05水平上显著，而胆汁盐浓度的显著性概率P FModel 2 0.00023670 0.00011835 1553.36 |t|Intercept 1 -0.03651 0.00484 -7.54 0.0001h 1 0.00031062 0.00004217 7.37 0.0002w 1 0.00072984 0.00004228 17.26 FModel 8 53.17231 6.64654 64.33 |t|Intercept 1 -6.14927 1.68879 -3.64 0.0010x1 1 -0.22296 0.20853 -1.07 0.2932x2 1 0.56813 0.25038 2.27 0.0304x3 1 0.01771 0.19207 0.09 0.9271x4 1 -0.12007 0.12562 -0.96 0.3466x5 1 -0.39754 0.31415 -1.27 0.2151x6 1 0.20935 0.19346 1.08 0.2875x7 1 -0.34198 0.23671 -1.44 0.1586x8 1 0.21464 0.20076 1.07 0.2932从参数估计列可以得到回归方程：复相关系数：（2）逐步回归分析：options linesize=76 nodate;data stepreg;infile e:dataer11-3e.dat;input y x1-x8;run;proc reg;model y=x1-x8/selection=stepwiseslentry=0.05 slstay=0.05;run;The SAS SystemThe REG ProcedureModel: MODEL1Dependent Variable: yStepwise Selection: Step 1Variable x2 Entered: R-Square = 0.9188 and C(p) = 8.2905Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr FModel 1 51.79923 51.79923 430.17 FIntercept -10.24579 0.64700 30.19713 250.78 .0001x2 0.39483 0.01904 51.79923 430.17 FModel 2 52.39734 26.19867 243.70 FIntercept -8.33902 1.01352 7.27767 67.70 .0001x2 0.41889 0.02068 44.11123 410.32 FModel 3 52.81516 17.60505 178.04 FIntercept -8.42672 0.97297 7.41726 75.01 .0001x2 0.35766 0.03579 9.87513 99.87 F1 x2 1 0.9188 0.9188 8.2905 430.17 FModel 1 39.96265 39.96265 276.71 FIntercept -14.87681 1.08114 27.34609 189.35 .0001x1 0.63413 0.03812 39.96265 276.71 FModel 2 40.61122 20.30561 162.40 FIntercept -13.31331 1.21786 14.94162 119.50 .0001x1 0.44066 0.09205 2.86530 22.92 FModel 3 41.12125 13.70708 125.03 FIntercept -13.50669 1.14392 15.28437 139.41 .0001x1 0.56648 0.10408 3.24772 29.62 F1 x1 1 0.9111 0.9111 11.3797 276.71 .00012 x6 2 0.0148 0.9259 7.3289 5.19 0.03123 x8 3 0.0116 0.9375 4.5706 4.65 0.0408在0.05水平上筛选出三个变量，它们分别是：X1，X6和X8。回归方程为：方差分析表：变差来源平方和自由度均方FP回 归41.121 25313.707 08125.03 FModel 2 35.61535 17.80767 15.61 0.0071Error 5 5.70225 1.14045Corrected Total 7 41.31760Root MSE 1.06792 R-Square 0.8620Dependent Mean 8.05000 Adj R-Sq 0.8068Coeff Var 13.26607Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr |t|Intercept 1 -19.57060 5.70767 -3.43 0.0187x1 1 7.29381 1.41032 5.17 0.0035x2 1 -0.44357 0.08239 -5.38 0.0030回归方程为：一次项和二次项的回归系数都是极显著的。11.6 两种农药“呋喃丹”和“铁灭克”，在不同 pH条件下对土壤磷酸酶活性（mg/g）的影响如下表所示81：缓冲液pH 呋喃丹(Y1)铁灭克(Y2)7.90.190.108.31.370.798.71.311.099.11.651.219.31.491.299.61.120.8710.01.070.7810.50.310.2211.00.120.10分别绘出呋喃丹和铁灭克对pH的散点图，计算出回归方程并求出磷酸酶活性达到最大值时的pH值，以及在该pH时磷酸酶的活性值。答：计算程序与上题一样，不再给出，只给出结果。（1）呋喃丹：The SAS SystemThe REG ProcedureModel: MODEL1Dependent Variable: y1Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr FModel 2 2.25859 1.12929 12.37 0.0074Error 6 0.54770 0.09128Corrected Total 8 2.80629Root MSE 0.30213 R-Square 0.8048Dependent Mean 0.95889 Adj R-Sq 0.7398Coeff Var 31.50847Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr |t|Intercept 1 -41.71019 9.89391 -4.22 0.0056x1 1 9.34395 2.10852 4.43 0.0044x2 1 -0.50595 0.11147 -4.54 0.0039回归方程为：一次项和二次项的回归系数都是极显著的。最大值的计算：1.011 9 X9.343 95 X9.234 06 Y1.431 13故当pH9.234 06时磷酸酶活性有最大值，其最大值为1.431 13。（2）铁灭克：The SAS SystemThe REG ProcedureModel: MODEL2Dependent Variable: y2Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr FModel 2 1.46564 0.73282 15.38 0.0044Error 6 0.28596 0.04766Corrected Total 8 1.75160Root MSE 0.21831 R-Square 0.8367Dependent Mean 0.71667 Adj R-Sq 0.7823Coeff Var 30.46194Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr |t|Intercept 1 -35.50332 7.14903 -4.97 0.0025x1 1 7.88157 1.52355 5.17 0.0021x2 1 -0.42419 0.08054 -5.27 0.0019回归方程为：一次项和二次项的回归系数都是极显著的。最大值的计算：0.848 38 X7.881 57 X9.290 14 Y1.107 13故当pH9.290 14时磷酸酶活性有最大值，其最大值为1.107 13。11.7 “武运粳7号”考种相关数据见下表82：序号产量/（kg hm-2）千粒重/g每穗总粒数/粒亩有效穗/（104hm-2）株高/cm19 787.525.9125.7372.30102.529 390.025.8131.3363.75105.639 607.526.3122.5370.8099.349 547.525.9128.3377.7098.959 237.026.5127.8358.65103.568 947.525.8137.5340.05100.378 277.525.7118.2372.9098.888 475.526.2113.6373.9597.698 415.025.9118.9373.0597.3108 040.025.4118.5356.7095.3118 167.526.1121.3333.6095.6127 845.025.3124.7345.7595.1137 927.525.8121.6343.5094.7147 327.525.6112.5343.2094.5157 305.025.9103.8362.4093.6167 125.025.4123.1319.2092.5177 140.026.1113.8308.5589.6186 945.026.4111.5306.4590.5以产量为因变量，计算多元回归方程，通过逐步回归筛选出对产量影响的重要因素。答：（1）多元回归方程见下表：The SAS SystemThe REG ProcedureModel: MODEL1Dependent Variable: yAnalysis of VarianceSum of MeanSource DF Squares Square F Value Pr FModel 4 14038414 3509603 50.79 |t|Intercept 1 -3